Mathematical Sciences: Fourier Analysis and Partial Differential Equations

在这一数学研究过程中，将涉及三个主要领域；几何，分析和经济建模。黎曼几何的Yamabe问题是问一个给定黎曼度规的流形是否与另一个常数曲率的流形保形等价，如果新的度规是由旧的度规乘以一个正函数得到的。本文将研究柯西-黎曼结构的类似问题。这里我们要选择一个具有常数伪厄米标量曲率的利瓦伊形式。这相当于在流形上解一个显式的偏微分方程。关于紧紧严格伪凸可定向流形的结果在奇维下得到了广泛的推广。其余案件的工作将继续进行。还将努力扩展基本(P, q1) -函数的幂积分和函数在热算符下的像之间的估计。过去二十年的结果虽然尖锐，但却是建立在两个指数是对偶的假设之上的。这些结果都符合Sobolev不等式的一般定义。对这种比较的新兴趣与偏微分方程解的唯一性有关。除了寻找新的Sobolev不等式外，还将把它们扩展到更一般的微分算子上。最近人们对经济学的兴趣源于弗罗贝纽斯关于微分方程的结论，用现代术语来说，微分方程说明了一种形式何时具有积分因子。在微观经济学中，人们考虑收入差异与收入和价格的需求函数（乘以价格差异）之间的一种形式的差异。这些形式区分了切向量是否指向改进的方向。Frobenius结果是效用函数存在的陈述——它通常被视为一个基本公理。有证据表明，这一公理不适用于消费者集合。本项目将侧重于描述可以预期近似可积性的条件。

Three major areas will be addressed during the course of this mathematical research; geometry, analysis and economic modeling. The Yamabe problem of Riemannian geometry asks if a manifold with a given Riemannian metric is conformally equivalent to another with constant scalar curvature if the new metric is obtained from the old by multiplication by a positive function. Work will be done on the analogous problem for Cauchy-Riemann structures. Here one seeks to choose a Levi form with constant pseudohermitian scalar curvature. This amounts to solving an explicit partial differential equation on the manifold. Results on compact strictly pseudoconvex orientable manifolds have been extensive for odd dimensions. Work will proceed on the remaining cases. Efforts will also be made to extend fundamental (P, q1) - estimates between power integrals of a function and the image of the function under the heat operator. Results over the past two decades, while sharp, have rested on the assumption that the two exponents are in duality. Such results all fall within the general definition of Sobolev inequalities. The renewed interest in such comparisons is related to work on uniqueness properties of solutions of partial differential equations. In addition to seeking new Sobolev inequalities work will also be done in expanding their applicability to more general differential operators. Recent interest in economics derives from a result of Frobenius about differential equations which, in modern terminology, states when a one-form has an integrating factor. In microeconomics one considers one-forms of differences between differentials of income and demand functions of income and prices (multiplied by differentials of prices). These forms distinguish whether or not a tangent vector is pointing in the direction of improvement. The Frobenius result is the statement that a utility function exists - it is generally regarded as a basic axiom. There is evidence to suggest that the axiom fails for collections of consumers. The present project will focus on characterizing conditions when approximate integrability can be expected.